A Numerical Study for a Cell System

In this paper we study an assemblage of N cells in which a chemical reaction takes place whose generation term is described by a function g(λ,c_j) where λ stands for a control parameter and c_j denotes the concentration of a substrate in the j-th cell. The cells communicate via membranes M (compare Fig.1) admitting diffusive transport with diffusion coefficients D_j. The j-th cell may be connected to an outside reservoir which feeds constant concentration α into the system through a membrane with the diffusion constant E_j. If a cell is not connected to the reservoir we put E_j=0. Next we introduce the new variable x_j=α-c_j (j = 1, ... N) and define the function f(λ, x)=g(λ, α-x). With this notation the steady states of our assemblage of cells are the solutions of the system (1a)$$({D_2} + {E_1}){x_1} - {D_2}{x_2} = f(\lambda ,{x_1})$$(1b)$$- {D_j}{x_{j - 1}} + ({D_j} + {D_{j + 1}} + {E_j}){x_j} - {D_{j + 1}}{x_{j + 1}} = f(\lambda ,{x_j})\;(j = 2, \ldots ,N - 1)$$(1c)$$- {D_N}{x_{N - 1}} + ({D_N} + {E_N}){x_N} = f(\gamma ,{x_N})$$.